Question

what are the lengths of the other two sides of the triangle? ac = 5 and bc = 5; ac = 5 and bc = 5√5; ac = 5√3 and bc = 5; ac = 5 and bc = 5√3

Explanation:

Step1: Identify triangle type

This is a 30 - 60 - 90 right - triangle. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

The hypotenuse \(AB = 10\). Since the hypotenuse is \(2x\), we set \(2x=10\), so \(x = 5\).

Step2: Find lengths of AC and BC

• The side opposite \(30^{\circ}\) (angle at B) is \(AC\). So \(AC=x = 5\).
• The side opposite \(60^{\circ}\) (angle at A) is \(BC\). So \(BC=x\sqrt{3}=5\sqrt{3}\).

Answer:

AC = 5 and BC = \(5\sqrt{3}\) (the last option: AC = 5 and BC = \(5\sqrt{3}\))